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%I #9 Nov 10 2013 09:15:21
%S 0,1,1,1,2,1,1,1,2,2,2,4,4,1,4,3,1,2,3,2,5,5,3,2,3,2,4,3,3,8,4,1,3,2,
%T 2,11,5,1,5,5,4,4,5,4,7,4,3,7,6,3,9,4,2,5,4,3,12,7,2,4,10,1,7,8,4,10,
%U 7,3,10,9,5,8,5,4,10,9,5,10,9,3,12,13,4,4,9,4,11,10,5,11,16,5,10,8,5,16,8,3,11,15
%N Number of ordered ways to write n = x + y (x, y > 0) with p(3, x) + p(6, y) prime, where p(3, k) denotes the triangular number k*(k+1)/2 and p(6, k) denotes the hexagonal number k*(2*k-1) = p(3, 2*k-1).
%C Conjecture: a(n) > 0 for all n > 1.
%C This implies that there are infinitely many primes each of which can be written as a sum of a triangular number and a hexagonal number.
%C See also A228425, A228428, A228429 and A228430 for more similar conjectures.
%H Zhi-Wei Sun, <a href="/A228431/b228431.txt">Table of n, a(n) for n = 1..10000</a>
%e a(14) = 1 since 14 = 10 + 4 with p(3, 10) + p(6, 4) = 83 prime.
%e a(38) = 1 since 38 = 31 + 7 with p(3, 31) + p (6, 7) = 587 prime.
%t p[m_,x_]:=(m-2)x(x-1)/2+x
%t a[n_]:=Sum[If[PrimeQ[p[3,x]+p[6,n-x]],1,0],{x,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000217, A000384, A228425, A228428, A228429, A228430.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Nov 10 2013
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